Intro to string theory g terhooft

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INTRODUCTION TO STRING THEORY ∗ version 14-05-04 Gerard ’t Hooft Institute for Theoretical Physics Utrecht University, Leuvenlaan 4 3584 CC Utrecht, the Netherlands and Spinoza Institute Postbox 80.195 3508 TD Utrecht, the Netherlands e-mail: g.thooft@phys.uu.nl internet: http://www.phys.uu.nl/~thooft/ Contents 1 Strings in QCD. 4 1.1 The linear trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 The Veneziano formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 The classical string. 7 3 Open and closed strings. 11 3.1 The Open string. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 The closed string. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3.1 The open string. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3.2 The closed string. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 The light-cone gauge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.5 Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.5.1 for open strings: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 ∗ Lecture notes 2003 and 2004 1 3.5.2 for closed strings: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.6 Energy, momentum, angular momentum. . . . . . . . . . . . . . . . . . . . 17 4 Quantization. 18 4.1 Commutation rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 The constraints in the quantum theory. . . . . . . . . . . . . . . . . . . . . 19 4.3 The Virasoro Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4 Quantization of the closed string . . . . . . . . . . . . . . . . . . . . . . . 23 4.5 The closed string spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5 Lorentz invariance. 25 6 Interactions and vertex operators. 27 7 BRST quantization. 31 8 The Polyakov path integral. Interactions with closed strings. 34 8.1 The energy-momentum tensor for the ghost fields. . . . . . . . . . . . . . . 36 9 T-Duality. 38 9.1 Compactifying closed string theory on a circle. . . . . . . . . . . . . . . . . 39 9.2 T-duality of closed strings. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 9.3 T-duality for open strings. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 9.4 Multiple branes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 9.5 Phase factors and non-coinciding D-branes. . . . . . . . . . . . . . . . . . 42 10 Complex coordinates. 43 11 Fermions in strings. 45 11.1 Spinning point particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 11.2 The fermionic Lagrangian. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 11.3 Boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 11.4 Anticommutation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 11.5 Spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 11.6 Supersymmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 11.7 The super current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2 11.8 The light-cone gauge for fermions . . . . . . . . . . . . . . . . . . . . . . . 56 12 The GSO Projection. 58 12.1 The open string. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 12.2 Computing the spectrum of states. . . . . . . . . . . . . . . . . . . . . . . 61 12.3 String types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 13 Zero modes 65 13.1 Field theories associated to the zero modes. . . . . . . . . . . . . . . . . . 68 13.2 Tensor fields and D-branes. . . . . . . . . . . . . . . . . . . . . . . . . . . 71 13.3 S-duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 14 Miscelaneous and Outlook. 75 14.1 String diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 14.2 Zero slope limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 14.2.1 Type II theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 14.2.2 Type I theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 14.2.3 The heterotic theories . . . . . . . . . . . . . . . . . . . . . . . . . 77 14.3 Strings on backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 14.4 Coordinates on D-branes. Matrix theory. . . . . . . . . . . . . . . . . . . . 78 14.5 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 14.6 Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 14.7 Black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 14.8 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3 1. Strings in QCD. 1.1. The linear trajectories. In the ’50’s, mesons and baryons were found to have many excited states, called res- onances, and in the ’60’s, their scattering amplitudes were found to be related to the so-called Regge trajectories: J = α(s), where J is the angular momentum and s = M 2 , the square of the energy in the center of mass frame. A resonance occurs at those s values where α(s) is a nonnegative integer (mesons) or a nonnegative integer plus 1 2 (baryons). The largest J values at given s formed the so-called ‘leading trajectory’. Experimentally, it was discovered that the leading trajectories were almost linear in s: α(s) = α(0) + α  s . (1.1) Furthermore, there were ‘daughter trajectories’: α(s) = α(0) − n + α  s . (1.2) where n appeared to be an integer. α(0) depends on the quantum numbers such as strangeness and baryon number, but α  appeared to be universal, approximately 1 GeV −2 . It took some time before the simple question was asked: suppose a meson consists of two quarks rotating around a center of mass. What force law could reproduce the simple behavior of Eq. (1.1)? Assume that the quarks move highly relativistically (which is reasonable, because most of the resonances are much heavier than the lightest, the pion). Let the distance between the quarks be r. Each has a transverse momentum p. Then, if we allow ourselves to ignore the energy of the force fields themselves (and put c = 1), s = M 2 = (2p) 2 . (1.3) The angular momentum is J = 2 p r 2 = p r . (1.4) The centripetal force must be F = p c r/2 = 2p r . (1.5) For the leading trajectory, at large s (so that α(0) can be ignored), we find: r = 2J √ s = 2α  √ s ; F = s 2 J = 1 2 α  , (1.6) or: the force is a constant, and the potential between two quarks is a linearly rising one. But it is not quite correct to ignore the energy of the force field, and, furthermore, the above argument does not explain the daughter trajectories. A more satisfactory model of the mesons is the vortex model: a narrow tube of field lines connects the two quarks. This 4 linelike structure carries all the energy. It indeed generates a force that is of a universal, constant strength: F = dE/dr. Although the quarks move relativistically, we now ignore their contribution to the energy (a small, negative value for α(0) will later be attributed to the quarks). A stationary vortex carries an energy T per unit of length, and we take this quantity as a constant of Nature. Assume this vortex, with the quarks at its end points, to rotate such that the end points move practically with the speed of light, c. At a point x between − r/2 and r/2, the angular velocity is v(x) = c x/(r/2). The total energy is then (putting c = 1): E =  r/2 −r/2 T dx √ 1 − v 2 = T r  1 0 (1 − x 2 ) −1/2 dx = 1 2 π T r , (1.7) while the angular momentum is J =  r/2 −r/2 T v x dx √ 1 − v 2 = 1 2 T r 2  1 0 x 2 dx √ 1 − x 2 = T r 2 π 8 . (1.8) Thus, in this model also, J E 2 = 1 2πT = α  ; α(0) = 0 , (1.9) but the force, or string tension, T , is a factor π smaller than in Eq. (1.6). 1.2. The Veneziano formula. 4 32 1 Consider elastic scattering of two mesons, (1) and (2), forming two other mesons (3) and (4). Elastic here means that no other particles are formed in the process. The ingoing 4-momenta are p (1) µ and p (2) µ . The outgoing 4-momenta are p (3) µ and p (4) µ . The c.m. energy squared is s = −(p (1) µ + p (2) µ ) 2 . (1.10) An independent kinematical variable is t = −(p (1) µ − p (4) µ ) 2 . (1.11) Similarly, one defines u = −(p (1) µ − p (3) µ ) 2 , (1.12) 5 but that is not independent: s + t + u = 4  i=1 m 2 (i) . (1.13) G. Veneziano asked the following question: What is the simplest model amplitude that shows poles where the resonances of Eqs. (1.1) and (1.2) are, either in the s-channel or in the t-channel? We do not need such poles in the u-channel since these are often forbidden by the quantum numbers, and we must avoid the occurrence of double poles. The Gamma function, Γ(x), has poles at negative integer values of x, or, x = 0, −1, −2, ···. Therefore, Veneziano tried the amplitude A(s, t ) = Γ(−α(s))Γ(−α(t)) Γ(−α(s) −α(t)) . (1.14) Here, the denominator was planted so as to avoid double poles when both α(s) and α(t) are nonnegative integers. This formula is physically acceptable only if the trajectories α(s) and α (t) are linear, for the following reason. Consider the residue of one of the poles in s. Using Γ(x) → (−1) n n! 1 x+n when x → −n, we see that α(s) → n ≥ 0 : A(s, t) → (−1) n n! 1 n − α(s) Γ(−α(t)) Γ(−α(t) −n) . (1.15) Here, the α(t) dependence is the p olynomial Γ( a + n ) / Γ( a ) = ( a + n − 1) ··· ( a + 1) a ; a = − α ( t ) − n , (1.16) called the Pochhammer polynomial. Only if α(t) is linear in t, this will be a polynomial of degree n in t. Notice that, in the c.m. frame, t = −(p (1) µ − p (4) µ ) 2 = m 2 (1) + m 2 (4) − 2E (1) E (4) + 2|p (1) ||p (4) |cos θ . (1.17) Here, θ is the scattering angle. In the case of a linear trajectory in t, we have a polynomial of degree n in cos θ. From group representation theory, we know that, therefore, the intermediate state is a superposition of states with angular momentum J maximally equal to n. We conclude that the n th resonance in the s channel consists of states whose angular momentum is maximally equal to n. So, the leading tra jectory has J = α(s), and there are daughter trajectories with lower angular momentum. Notice that this would not be true if we had forgotten to put the denominator in Eq. (1.14), or if the trajectory in t were not linear. Since the Pochhammer polynomials are not the same as the Legendre polynomials, superimposed resonances appear with J lower than n, the daughters. An important question concerns the sign of these contributions. A negative sign could indicate intermediate states with indefinite metric, which would be physically unrealistic. In the early ’70s, such questions were investigated purely mathematically. Presently, we know that it is more fruitful to study the physical interpretation of Veneziano’s amplitude (as well as generalizations thereof, which were soon discovered). 6 The Veneziano amplitude A(s, t) of Eq. (1.14) is the beta function: A(s, t ) = B(−α(s), −α(t)) =  1 0 x −α(s)−1 (1 − x) −α(t)−1 dx . (1.18) The fact that the poles of this amplitude, at the leading values of the angular momen- tum, obey exactly the same energy-angular momentum relation as the rotating string of Eq. (1.9), is no coincidence, as will be seen in what follows (section 6, Eq. (6.22)). 2. The classical string. Consider a kind of material that is linelike, being evenly distributed over a line. Let it have a tension force T . If we stretch this material, the energy we add to it is exactly T per unit of length. Assume that this is the only way to add energy to it. This is typical for a vortex line of a field. Then, if the material is at rest, it carries a mass that (up to a factor c 2 , which we put equal to one) is also T per unit of length. In the simplest conceivable case, there is no further structure in this string. It then does not alter if we Lorentz transform it in the longitudinal direction. So, we assume that the energy contained in the string only depends on its velocity in the transverse direction. This dependence is dictated by relativity theory: if u µ ⊥ is the 4-velocity in the transverse direction, and if both the 4-momentum density p µ and u µ transform the same way under transverse Lorentz transformations, then the energy density dU/d must be just like the energy of a particle in 2+1 dimensions, or dU d = T  1 − v 2 ⊥ /c 2 . (2.1) In a region where the transverse velocity v ⊥ is non-relativistic, this simply reads as U = U kin + V ; U kin =  1 2 T v 2 ⊥ d , V =  T d , (2.2) which is exactly the energy of a non-relativistic string with mass density T and a tension T , responsible for the potential energy. Indeed, if we have a string stretching in the z -direction, with a tiny deviation ˜x(z), where ˜x is a vector in the (xy)-direction, then d dz =  1 +  ∂˜x ∂z  2 ≈ 1 + 1 2  ∂˜x ∂z  2 ; (2.3) U ≈  dz  T + 1 2 T  ∂˜x ∂z  2 + 1 2 T ( ˙ ˜x) 2  . (2.4) We recognize a ‘field theory’ for a two-component scalar field in one space-, one time- dimension. In the non-relativistic case, the Lagrangian is then L = U kin − V = −  T (1 − 1 2 v 2 ⊥ ) d = −  T  1 − v 2 ⊥ d . (2.5) 7 Since the eigen time dτ for a point moving in the transverse direction along with the string, is given by dt  1 − v 2 ⊥ , we can write the action S as S =  L dt = −  T d dτ . (2.6) Now observe that this expression is Lorentz covariant. Therefore, if it holds for describing the motion of a piece of string in a frame where it is non-relativistic, it must describe the same motion in all lorentz frames. Therefore, this is the action of a string. The ‘surface element’ d dτ is the covariant measure of a piece of a 2-surface in Minkowski space. To understand hadronic particles as excited states of strings, we have to study the dynamical properties of these strings, and then quantize the theory. At first sight, this seems to be straightforward. We have a string with mass per unit of length T and a tension force which is also T (in units where c = 1). Think of an infinite string stretching in the z direction. The transverse excitation is described by a vector x tr (z, t) in the x y direction, and the excitations move with the speed of sound, here equal to the speed of light, in the positive and negative z -direction. This is nothing but a two-component massless field theory in one space-, one time-dimension. Quantizing that should not be a problem. Yet it is a non-linear field theory; if the string is strongly excited, it no longer stretches in the z -direction, and other tiny excitations then move in the z-direction slower. Strings could indeed reorient themselves in any direction; to handle that case, a more powerful scheme is needed. This would have been a hopeless task, if a fortunate accident would not have occurred: the classical theory is exactly soluble. But, as we shall see, the quantization of that exact solution is much more involved than just a renormalizable massless field theory. In Minkowski space-time, a string sweeps out a 2-dimensional surface called the “world sheet”. Introduce two coordinates to describe this sheet: σ is a coordinate along the string, and τ a timelike coordinate. The world sheet is described by the functions X µ (σ, τ), where µ runs from 0 to d, the number of space dimensions 1 . We could put τ = X 0 = t, but we don’t have to. The surface element d dτ of Eq. (2.6) will in general be the absolute value of Σ µν = ∂ X µ ∂ σ ∂ X ν ∂ τ − ∂ X ν ∂ σ ∂ X µ ∂ τ , (2.7) We have 1 2 Σ µν Σ µν = (∂ σ X µ ) 2 (∂ τ X ν ) 2 − (∂ σ X µ ∂ τ X µ ) 2 . (2.8) The surface element on the world sheet of a string is timelike. Note that we are assuming the sign convention (− + + +) for the Minkowski metric; throughout these notes, a repeated index from the middle of the Greek alphabet is read as follows: X µ 2 ≡ η µν X µ X ν = X 1 2 + X 2 2 + ··· + (X D−1 ) 2 − X 0 2 , 1 We use D to denote the total number of spacetime dimensions: d = D − 1. 8 where D stands for the number of space-time dimensions, usually (but not always) D = 4. We must write the Lorentz invariant timelike surface element that figures in the action as S = −T  dσ dτ  (∂ σ X µ ∂ τ X µ ) 2 − (∂ σ X µ ) 2 (∂ τ X ν ) 2 . (2.9) This action, Eq. (2.9), is called the Nambu-Goto action. One way to proceed now is to take the coordinates σ and τ to be light-cone coordinates on the string world sheet. In order to avoid confusion later, we refer to such coordinates as σ + and σ − instead of σ and τ . These coordinates are defined in such a way that (∂ + X µ ) 2 = (∂ − X µ ) 2 = 0 . (2.10) The second term inside the square root is then a double zero, which implies that it also vanishes to lowest order if we consider an infinitesimal variation of the variables X µ (σ + , σ − ). Thus, keeping the constraint (2.10) in mind, we can use as our action S = T  ∂ + X µ ∂ − X µ dσ + dσ − . (2.11) With this action being a bilinear one, the associated Euler-Lagrange equations are linear, and easy to solve: ∂ + ∂ − X µ = 0 ; X µ = a µ (σ + ) + b µ (σ − ) . (2.12) The conditions (2.10) simply imply that the functions a µ (σ + ) and b µ (σ − ), which would otherwise be arbitrary, now have to satisfy one constraint each: (∂ + a µ (σ + )) 2 = 0 ; (∂ − b µ (σ − )) 2 = 0 . (2.13) It is not hard to solve these equations: since ∂ + a 0 =  (∂ + a) 2 , we have a 0 (σ + ) =  σ +  (∂ + a(σ 1 )) 2 dσ 1 ; b 0 (σ − ) =  σ −  (∂ +  b(σ 1 )) 2 dσ 1 , (2.14) which gives us a 0 (σ + ) and b 0 (σ − ), given a(σ + ) and  b(σ − ). This completes the classical solution of the string equations. Note that Eq. (2.11) can only be used if the sign of this quantity remains the same everywhere. Exercise: Show that ∂ + X µ ∂ − X µ can switch sign only at a point (σ + 0 , σ − 0 ) where ∂ + a µ (σ + 0 ) = C · ∂ − b µ (σ − 0 ). In a generic case, such points will not exist. This justifies our sign assumption. For future use, we define the induced metric h αβ (σ, τ) as h αβ = ∂ α X µ ∂ β X µ , (2.15) 9 where indices at the beginning of the Greek alphabet, running from 1 to 2, refer to the two world sheet coordinates, for instance: σ 1 = σ , σ 2 = τ , or, as the case may be, σ 1,2 = σ ± , (2.16) the distances between points on the string world sheet being defined by ds 2 = h αβ dσ α dσ β . The Nambu-Goto action is then S = −T  d 2 σ √ h ; h = −det αβ (h αβ ) , d 2 σ = dσ dτ . (2.17) We can actually treat h αβ as an independent variable when we replace the action (2.9) by the so-called Polyakov action: S = − T 2  d 2 σ √ h h αβ ∂ α X µ ∂ β X µ , (2.18) where, of course, h αβ stands for the inverse of h αβ . Varying this action with respect to h αβ gives h αβ → h αβ + δh αβ ; δS = T  d 2 σ δh αβ √ h (∂ α X µ ∂ β X µ − 1 2 h αβ h γδ ∂ γ X µ ∂ δ X µ ) . (2.19) Requiring δS in Eq. (2.19) to vanish for all δh αβ (σ, τ) does not give Eq. (2.15), but instead h αβ = C(σ, τ)∂ α X µ ∂ β X µ . (2.20) Notice, however, that the conformal factor C(σ, τ) cancels out in Eq. (2.18), so that varying it with respect to X µ (σ, τ) still gives the correct string equations. C is not fixed by the Euler-Lagrange equations at all. So-far, all our equations were invariant under coordinate redefinitions for σ and τ . In any two-dimensional surface with a metric h αβ , one can rearrange the coordinates such that h 12 = h 21 = 0 ; h 11 = −h 22 , or: h αβ = η αβ e φ , (2.21) where η αβ is the flat Minkowski metric diag(−1, 1) on the surface, and e φ some conformal factor. Since this factor cancels out in Eq. (2.18), the action in this gauge is the bilinear expression S = − 1 2 T  d 2 σ (∂ α X µ ) 2 . (2.22) Notice that in the light-cone coordinates σ ± = 1 √ 2 (τ ±σ), where the flat metric η αβ takes the form η αβ = −  0 1 1 0  , (2.23) 10 [...]... the gauge group: [Λa , Λb ](A) = f abc Λc (A) (7.24) ε is infinitesimal Eq (7.22) is in fact a gauge transformation generated by the infinites¯ imal field εη ¯ 8 The Polyakov path integral Interactions with closed strings Two closed strings can meet at one point, where they rearrange to form a single closed string, which later again splits into two closed strings This whole process can be seen as a single... multiple strings) This single string has a center of mass described by a wave function in space and time, using all D operators xµ (with p µ being the canonically associated operators −iη µν ∂/∂xν ) Then we have the string excitations The non-excited string mode is usually referred to as the ‘vacuum state’ | 0 (not to be confused with the spacetime vacuum, where no string is present at all) All string excited... converge In fact, we are not interested in doing the integrals along such orbits, we only want to integrate over states which are physically distinct This is why one needs to fix the gauge The simplest way to fix the gauge is by imposing a constraint on the field configurations Suppose that the set of infinitesimal gauge transformations is described by ‘generators’ Λa (x), where the index a can take a number... may also take place Strings could split and rejoin several times, in a process that would be analogous to a multi-loop Feynman diagram in Quantum Field Theory The associated string world sheets then take the form of a torus or sheets with more complicated topology: there could be g splittings and rejoinings, and the associated world sheet is found to be a closed surface of genus g The Polyakov action... Tµν 8.1 The energy-momentum tensor for the ghost fields gh We shall now go through the calculation of the ghost energy-momentum tensor Tαβ a bit more carefully than in Green-Schwarz-Witten, page 127 Rewrite the ghost part of the Lagrangian (8.14) as T ? Sgh = − 2 √ d2 σ h hαβ ( αc γ )bβγ , (8.16) where, by partial integration, we brought Eq (3.1.31) of Green-Schwarz-Witten in a slightly more convenient... edges, so that there is no force acting on them: the edges are free end points 3.2 The closed string In the case of a closed string, we choose as our boundary condition: X µ (σ, τ ) = X µ (σ + π, τ ) (3.4) Again, we must use transformations of the form (3.1) to guarantee that this condition is kept after fixing the conformal gauge The period π is in accordance with the usual convention in string theory. .. the energy per unit of length is P0 = ∂p0 ∂X 0 =T =T C∂σ C∂τ (3.42) Quite generally, one has ∂X µ (3.43) ∂τ Although this reasoning would be conceptually easier to understand if we imposed a “time gauge”, X 0 = Const · τ , all remains the same in the light-cone gauge In chapter 4, subsection 4.1 , we derive the energy-momentum density more precisely from the Lagrange formalism Pµ=T We have µ Ptot =... values (in YM theories: the dimensionality of the gauge group; in gravity: the dimensionality D of space-time, in string theory: 2 for the two dimensions of the string world sheet, plus one for the Weyl invariance) One chooses functions fa (x), such that the condition fa (x) = 0 (7.3) fixes the choice of gauge — assuming that all configurations can be gauge transformed such that this condition is obeyed... index of the gauge generators In perturbation expansion, we assume fa (x) at first order to be a linear function of the fields Ai (x) (and possibly its derivatives) Also the gauge transformation is linear at lowest order: ˆ Ai → Ai + Tia Λa (x) , (7.4) ˆ where Λa (x) is the generator of infinitesimal gauge transformations and Tia may be an operator containing partial derivatives If the gauge transformations... operators that generate a Lorentz boost Together, they form the tensor J µν that we derived in Eq (3.48) The string states, with all their properties that we derived, should be a representation of the Lorentz group What this means is the following If we compute the commutators of the operators (3.48), we should get the same operators at the right hand side as what is dictated by group theory: [ p µ, . the Nambu-Goto action. One way to proceed now is to take the coordinates σ and τ to be light-cone coordinates on the string world sheet. In order to avoid. following form. There is a single (open or closed) string (at a later stage, one might compose states with multiple strings). This single string has a